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Unfortunately this is impossible to predict.
I have seen the M to KS trail open in early June for a couple of days only to close again with a late snowstorm. Typically the trail opens around June 15. Same for the Eiger Trail.
The gondola from Wengen to Männlichen will be running by then and even if the trail is closed it is well worth the trip up to take in the fabulous views (if no clouds, of course)
Advice for early June:
there are a number of lower elevation hikes that are usually open during April, May, and early June. These are all easy walks; the more challenging hikes are those higher up.
A beautiful valley walk begins in Lauterbrunnen and ends in Stechelberg (or earlier, depending on your inclination). It is about 4 miles long on a wide flat path. You can go as far as Trümmelbach Falls (about half way and highly recommended for a visit) then bus the rest of the way, go as far as the Stechelberg Schilthornbahn station, about 3.5 miles total, or go all the way to Stechelberg where there is a lovely inn that serves delicious lunches. You can take the bus back from any of these stops to Lauterbrunnen.
The only walk at elevation that’s pretty sure to be open is the Grütschalp to Mürren route. Take the cable train from Lauterbrunnen to Grütschalp then enjoy the 3.5 mile, mostly flat stroll to Mürren. Stunning views of the Eiger massif from here. You can also walk from Mürren down the paved easy trail to Gimmelwald then catch the gondola down to Stechelberg.
The Allmenhubel lift (from Mürren up to Allmendhubel) generally opens June 11 this year. Walks in that area may be open that early or maybe not, entirely depending on snow levels.